330 Chapter 8. Sorting
For small N one does better to use an algorithm whose operation count goes
as a higher, i.e., poorer, power of N , if the constant in front is small enough. For
N < 20, roughly, the method of straight insertion (§8.1) is concise and fast enough.
We include it with some trepidation: It is an N 2 algorithm, whose potential for
misuse (by using it for too large an N ) is great. The resultant waste of computer
time is so awesome, that we were tempted not to include any N 2 routine at all. We
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will draw the line, however, at the inefﬁcient N 2 algorithm, beloved of elementary
computer science texts, called bubble sort. If you know what bubble sort is, wipe it
from your mind; if you don’t know, make a point of never ﬁnding out!
For N < 50, roughly, Shell’s method (§8.1), only slightly more complicated to
program than straight insertion, is competitive with the more complicated Quicksort
on many machines. This method goes as N 3/2 in the worst case, but is usually faster.
See references [1,2] for further information on the subject of sorting, and for
detailed references to the literature.
CITED REFERENCES AND FURTHER READING:
Knuth, D.E. 1973, Sorting and Searching, vol. 3 of The Art of Computer Programming (Reading,
MA: Addison-Wesley). [1]
Sedgewick, R. 1988, Algorithms, 2nd ed. (Reading, MA: Addison-Wesley), Chapters 8–13. [2]
8.1 Straight Insertion and Shell’s Method
Straight insertion is an N 2 routine, and should be used only for small N ,
say < 20.
The technique is exactly the one used by experienced card players to sort their
cards: Pick out the second card and put it in order with respect to the ﬁrst; then pick
out the third card and insert it into the sequence among the ﬁrst two; and so on until
the last card has been picked out and inserted.
void piksrt(int n, float arr[])
Sorts an array arr[1..n] into ascending numerical order, by straight insertion. n is input; arr
is replaced on output by its sorted rearrangement.
{
int i,j;
float a;
for (j=2;j 0 && arr[i] > a) { Look for the place to insert it.
arr[i+1]=arr[i];
i--;
}
arr[i+1]=a; Insert it.
}
}
What if you also want to rearrange an array brr at the same time as you sort
arr? Simply move an element of brr whenever you move an element of arr:

8.1 Straight Insertion and Shell’s Method 331
void piksr2(int n, float arr[], float brr[])
Sorts an array arr[1..n] into ascending numerical order, by straight insertion, while making
the corresponding rearrangement of the array brr[1..n].
{
int i,j;
float a,b;
for (j=2;j 0 && arr[i] > a) { Look for the place to insert it.
arr[i+1]=arr[i];
brr[i+1]=brr[i];
i--;
}
arr[i+1]=a; Insert it.
brr[i+1]=b;
}
}
For the case of rearranging a larger number of arrays by sorting on one of
them, see §8.4.
Shell’s Method
This is actually a variant on straight insertion, but a very powerful variant indeed.
The rough idea, e.g., for the case of sorting 16 numbers n1 . . . n16 , is this: First sort,
by straight insertion, each of the 8 groups of 2 (n1 , n9 ), (n2 , n10), . . . , (n8 , n16).
Next, sort each of the 4 groups of 4 (n1 , n5 , n9 , n13 ), . . . , (n4 , n8 , n12 , n16). Next
sort the 2 groups of 8 records, beginning with (n1 , n3 , n5 , n7 , n9 , n11, n13 , n15).
Finally, sort the whole list of 16 numbers.
Of course, only the last sort is necessary for putting the numbers into order. So
what is the purpose of the previous partial sorts? The answer is that the previous
sorts allow numbers efﬁciently to ﬁlter up or down to positions close to their ﬁnal
resting places. Therefore, the straight insertion passes on the ﬁnal sort rarely have to
go past more than a “few” elements before ﬁnding the right place. (Think of sorting
a hand of cards that are already almost in order.)
The spacings between the numbers sorted on each pass through the data (8,4,2,1
in the above example) are called the increments, and a Shell sort is sometimes
called a diminishing increment sort. There has been a lot of research into how to
choose a good set of increments, but the optimum choice is not known. The set
. . . , 8, 4, 2, 1 is in fact not a good choice, especially for N a power of 2. A much
better choice is the sequence
(3k − 1)/2, . . . , 40, 13, 4, 1 (8.1.1)
which can be generated by the recurrence
i1 = 1, ik+1 = 3ik + 1, k = 1, 2, . . . (8.1.2)
It can be shown (see [1]) that for this sequence of increments the number of operations
required in all is of order N 3/2 for the worst possible ordering of the original data.

332 Chapter 8. Sorting
For “randomly” ordered data, the operations count goes approximately as N 1.25, at
least for N < 60000. For N > 50, however, Quicksort is generally faster. The
program follows:
void shell(unsigned long n, float a[])
Sorts an array a[1..n] into ascending numerical order by Shell’s method (diminishing increment
sort). n is input; a is replaced on output by its sorted rearrangement.
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Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)
{
unsigned long i,j,inc;
float v;
inc=1; Determine the starting increment.
do {
inc *= 3;
inc++;
} while (inc a, and then scan another pointer down from the end of the array
until you ﬁnd an element < a. These two elements are clearly out of place for the
ﬁnal partitioned array, so exchange them. Continue this process until the pointers
cross. This is the right place to insert a, and that round of partitioning is done. The